This project will investigate the arithmetic and geometry encoded in the modular varieties arising from the study of automorphic forms, as well as in certain Calabi-Yau manifolds. The first part of the project will investigate the possibility of proving the Tate conjecture for the entire class of quaternionic Shimura surfaces, which if successful, will lead to progress on this question for divisors on all Shimura varieties of classical type. The second project, joint with K. Paranjape, will aim to associate Calabi-Yau varieties with involution over the rationals of dimension m to certain basic holomorphic cusp forms of weight m+1 and rational coefficients. The emphasis here will be on the 3-dimensional case, with a view to understanding quadratic twists and functorial products. The PI will also continue his investigation into certain other topics, including the works with N. Dunfield on the circle fibrations of hyperbolic 3-manifolds of arithmetic type, and with P. Michel concerning the exact averages of L-values.
Many problems one encounters become amenable to elucidation by the mathematical method when they exhibit some symmetry such as periodicity or invariance under mirror reflection. This is important in the cracking of codes and in the study of crystals and precious stones, for example. The main thrust of this project is to comprehend some of the manifestations of symmetry, especially when their presence is not evident. Mathematicians, Physicists, and others often start with discrete collections of numbers, possibly from experimentation, then form their generating functions, and ask if they encode hidden symmetries. When such harmonious arrangements arise in nature, they frequently describe the tones of automorphic functions, which are continuous entities like the waveforms on a disk. Their discrete frequencies are linked to exciting constructs like lengths of curves, prime numbers, and congruence solutions of polynomial equations. A particular focus of the project is to determine when the presence of Galois symmetries implies the existence of special curved subspaces of the ambient space, relating in turn to the poles of certain zeta functions. The ultimate aim is to understand the ubiquity and power of number formations better through geometry and analysis.
Intellectual Merit: A main focus of number theory is Diophantine equations: polynomial equations (those with two or more variables) in which the coefficients are all integers. In a joint work with M. Dimitrov, funded by this project, the PI has shown that a specific collection of systems of homogeneous equations with six variables has only a finite number of rational solutions up to scaling (which is a way to trivially modify and get new solutions). They identify the solutions with rational points on certain hyperbolic spaces of dimension 4 with complex structure. A simple example of such a hyperbolic complex surface is given by the simultaneous solutions of the equations: x15 + y5 = z5; x25 + w5 = z5; and x35 + w5 = y5, but in this case the finiteness was known before. Broader impact: Solving such diophantine equations has applications in everyday life, from balancing chemical equations to encryption. Every time one visits a website with an https:// address, it is likely that the website browser is using an encryption system that validates the certificate for the remote server to which one is trying to connect. The security keys that are exchanged point to a number-theoretic solution. One wants integer equations that are hard to solve without the key, and this is where number theory comes in. This project and related work involved several students at Caltech, both graduate and undergraduate. The PI also helped the diversity efforts at Caltech, spoke at local community colleges, ran seminars attended by students from a wide area, wrote up and posted (widely used) lecture notes online, edited books, co-authored a graduate text, and built bridges with other sciences by collaborating with a Chemist on the mathematics behind enzyme kinetics.
